Math 1220, Fall 2014
Homework due on September 11, 2014
You must type up one problem from this assignment in LaTeX. You may only
use another piece of software with Ravis permission.
From the text: 1.56, 2.1, 2.4, 2.5, 2.6, 2.7
Also,
1) Give an example of:
Math 1220, solutions to homework 3
Daniel Miller
September 17, 2013
1.46
a) By assumption we have a2k+1
a2k+1 0.
a2k+1 , which implies a2k+1 a2k+2
a2k+2 and a2k
b) We have s2(n+1) = s2n + (a2k+1 a2k+2 ) s2n and s2n+1 = s2(n1)+1 + (a2n + a2n+1 )
other word
Math 1220, solutions to homework 4
Daniel Miller
September 24, 2013
2.23 We need to show that (f g )(V ) = V and (g f )(H ) = H . But f (g (V ) is the volume of water
when the height is g (V ). But the height is g (V ) when the volume is V , so f (g (V )
Math 1220, solutions to homework 5
Daniel Miller
September 27, 2013
3.6
a) The volume between depths a and a + h.
b) We use the fact that A is decreasing at a. The quantity A(a)h is the volume of a cylinder of height
h whose base has area A(a), while A(a
Math 1220, solutions to homework 6
Daniel Miller
October 8, 2013
4.17 Let f (x) = h(x) g (x). Then f (x)
0 for all x > 0 and f (0) = 0. By Corollary 4.2, f is
nondecreasing on [0, ), which means that for all x 0, f (x) f (0) = 0, so f 0.
4.18
a) We have g
11.1 Calculate the variance of the numerical value when one die is
rolled.
1
Obviously the expected value of the die roll is 6 (1+2+3+4+5+6) =
3.5. The expected value of the squared dierence between 3.5 and the
35
value rolled is: 1 (2.52 + 1.52 + .52 + .
8.1a Compute the values of Ilef t , Iright and Imid in the following
cases: f (x) = x3 , [1, 2], n = 1, 2, 4
Ilef t = 2, 4.375, 5.84375
Imid = 6.75, 7.3125, 7.45313
Iright = 16, 11.375, 9.34375
1
2
8.2 When a drug is administered once every 24 hours, the
9.8 Verify that the polar representation for the reciprocal of z =
r(cos + i sin ) is z 1 = r1 (cos i sin ).
It suces to show that:
r(cos + i sin ) r1 (cos i sin ) = 1
Which is just
rr1 (cos + i sin )(cos i sin ) = (cos2 + sin2 ) = 1
1
2
9.9 Prove that fo
9.1 For each of the numbers z = 2 + 3i, z = 4 i, calculate the
following:
(1) Find |z |. We have 22 + 32 = 13 and 42 + (1)2 = 17
respectively
(2) Find z . These are 2 3i and 4 + i respectively
(3) Verify that z + z = 2Re (z ).
(2 + 3i) + (2 3i) = 4 = 2(2)
7.12 Give an example of asymptotic sequences an and bn , i.e. sen
quences for which the ratio an tends to 1. Does the dierence an bn
b
tend to 0 in your example? Must the dierence tend to 0?
1
an = n and bn = n + n are asymptotic, and their dierence does
6.20 When a spring is stretched or compressed a distance x, the
force required is kx, where k is a constant, and x is measured from
the equilibrium length. A spring requires a force of 2000 Newtons to
compress it 4 millimeters. Verify that the spring cons
6.1 Find a better estimate for the mass of the rod R(x, [1, 5]) discussed in section 6.1b, by
(1) subdividing the rod into four subpieces of equal length,
1(1) + 2(1) + 3(1) + 4(1)
R(x, [1, 2]) + R(x, [2, 3]) + R(x, [3, 4]) + R(x, [4, 5])
2(1) + 3(1) + 4(
4.14 Suppose g (x) h (x) for 0 < x and g (0) = h(0). Prove that
g (x) h(x) for 0 < x.
Let f (x) = h(x) g (x). Then we are given that f (x) 0 for
x > 0 and f (0) = 0. Thus f is increasing for x > 0, so if x > 0,
f (x) > f (0) = 0. So h(x) g (x) 0, so h(x)
3.8 The volume of water in a bottle is a function of the depth of the
water. Let V (a) be the volume up to depth a. Let A(a) be the area
of water exposed to the air, the cross sectional area of the bottle at
height a.
(1) What does V (a + h) V (a) represe
Math 1220, solutions to homework 2
Daniel Miller
September 10, 2013
1.25
No solution included.
1.26
No solution included.
1.29
Assuming s <
1.32
If we carelessly commute sums and limits, we nd
2, we have
1 = lim
n
1+
2
s
=
2
n
2
2
s
<
21=
2.
= lim (an + b
Math 1220: honors calculus II
Daniel Miller
October 11, 2013
Problems assigned
Chapter 6: 15, 16, 18, 21, 26, 27, 28, 30, 31, 32
Practice problems
6.17
Use the fundamental theorem of calculus to calculate the integrals.
/4
a)
0
dx
1 + x2
1
(x2 + 2)2 dx
b
Math 1220, Fall 2014
Homework due on September 15, 2014
You must type up one problem from this assignment in LaTeX. You may only
use another piece of software with Ravis permission.
From the text: 2.8, 2.10, 2.13, 2.16
Also,
1) Give an example of a bounde
Math 1220, Fall 2014
Homework due on September 8, 2014
Remember - I dont expect that everyone will solve every problem, but I do
expect that everyone make a serious attempt at every problem and explain what
you tried when you cant solve a problem.
From th
Math 1220, Fall 2014
Homework due on August 28, 2014
Remember - I dont expect that everyone will solve every problem, but I do
expect that everyone make a serious attempt at every problem and explain what
you tried when you cant solve a problem.
From the
Math 1220, Fall 2014
Homework due on September 4, 2014
Remember - I dont expect that everyone will solve every problem, but I do
expect that everyone make a serious attempt at every problem and explain what
you tried when you cant solve a problem.
From th
Math 1220, Fall 2014
Homework due on September 2, 2014
Remember - I dont expect that everyone will solve every problem, but I do
expect that everyone make a serious attempt at every problem and explain what
you tried when you cant solve a problem.
From th
Math 1220: honors calculus II
Daniel Miller
August 30, 2013
Problems assigned
Chapter 1: 2, 5, 7, 8, 12, 15, 16, 19, 21, 23, 24.
Practice problems
1.1
Find the numbers that satisfy each inequality, and sketch the solution on a number line.
(a) |x 3|
4
(b)
Math 1220: honors calculus II
Daniel Miller
September 13, 2013
Problems assigned
Chapter 1: 46, 48, 50, 51, 52, 54, Chapter 2: 1, 2, 7, 11 (extra credit), 14, 15, 16 17
Some induction
Try proving the following using induction:
n
3
i=0 i
a)
For all n
1,
b)
Math 1220: honors calculus II
Daniel Miller
September 20, 2013
Problems assigned
Chapter 2: 23, 25, 30, 40, 41, 57, 59, 61, 62, 63, 64, 70, 71.
On proof by example
1 Show that for all functions f we have f = f .
2 Show that xn + y n = z n has integer solu
PRELIM 1 REVIEW LIST
Chapter 1 Denitions to know, what they mean and to know how to use
1. Geometric sequence, geometric series.
2. Sequence, series of numbers.
3. Sequence convergence.
4. How to interpret |x a| < geometrically,i.e. as an interval; comput
Math 1220: honors calculus II
Daniel Miller
September 27, 2013
Problems assigned
Chapter 3: 6, 9, 10, 44, 56, 57, 67, 69, 70
Practice problems
3.13
Let g (x) = x2 sin
x
, g (0) = 0.
a) Sketch the graph of g on [1, 1].
b) Show that g is continuous at 0.
c)
MATH 1220 PRELIM 1 October 1, 2013
This is a 75 minute test. No notes or calculators are allowed.
There are 6 questions. See both sides. Please write your answers in
the test booklet, clearly numbering your answers to each question.
Show all your work. An